Navier–Stokes Transport Coefficients for Multicomponent Granular Gases. I. Theoretical Results

Abstract

The Chapman–Enskog method is applied to solve the set of Enskog kinetic equations of a multicomponent mixture of smooth inelastic hard spheres. As with monocomponent systems, an analysis is performed to first-order in spatial gradients. The Navier–Stokes transport coefficients and the first-order contribution to the cooling rate are obtained in terms of the solution to a set of coupled linear integral equations. These equations are approximately solved by considering the leading terms in a Sonine polynomial expansion. Explicit forms of the relevant transport coefficients of the mixture are obtained in terms of concentrations, masses and sizes of the constituents of the mixture, solid volume fraction, and coefficients of restitution. The dependence of these coefficients on the parameter space of the system is amply illustrated in the case of a binary mixture.

Appendices

Appendix A

The explicit form of the transport coefficients associated with mass and heat fluxes requires knowledge of the quantities $I_{ij\ell }$. These parameters are defined in terms of the functional derivative of the local pair distribution function $\chi _{ij}$ with respect to the local partial densities $n_\ell$ [15]:

$$\begin{aligned} I_{ij\ell }(\sigma _{ij};\left\{ n_{k}\right\} )=&\frac{2 n_{\ell }(\mathbf {r}_{1},t)}{\chi _{ij}\left( \sigma _{ij};\left\{ n_{k}\right\} \right) \sigma _{ij}}\int \text {d}\mathbf {r}' \left( \widehat{\varvec{\sigma }}_{ij}\cdot \mathbf {r}'\right) \nonumber \\&\times \left| \frac{\delta \chi _{ij}\left( -\frac{1}{2}{\varvec{\sigma }}_{ij},+\frac{1}{2}{\varvec{\sigma }}_{ij}\mid \left\{ n_{i}\right\} \right) }{\delta n_{\ell }(\mathbf {r}',t)}\right| _{\delta n=0}. \end{aligned}$$

(5.97)

These quantities are zero if $i=j$; they are nonzero otherwise. As mentioned in the main text, they are chosen here to recover the results obtained for ordinary fluid mixtures [8], given the mathematical difficulties involved in their direct evaluation. Thus, the parameters $I_{i \ell j}$ are defined through the relation [33]

$$\begin{aligned} \sum _{\ell =1}^2 n_\ell \sigma _{i\ell }^d \chi _{i\ell }\left( n_j \frac{\partial \ln \chi _{i\ell }}{\partial n_j}+I_{i\ell j}\right) =&\frac{d \varGamma \left( \frac{d}{2}\right) }{\pi ^{d/2}}\left[ \frac{n_j}{T}\left( \frac{\partial \mu _i}{\partial n_j}\right) _{T,n_{k\ne j}}-\delta _{ij}\right] \nonumber \\&-2n_j\chi _{ij}\sigma _{ij}^d, \end{aligned}$$

(5.98)

where $\mu _i$ is the chemical potential of species i [30]. Since granular fluids lack a thermodynamic description (they are inherently out of equilibrium), the concept of chemical potential is questionable. Here, for practical purposes, the expression of $\mu _i$ obtained for ordinary mixtures will be taken. Although this evaluation requires the use of thermodynamic relations that only hold for ordinary systems, it is expected that this approximation will be reliable for not too strong values of dissipation. This assertion must of course be supported by computer simulations.

In a binary mixture, the relevant nonzero parameters are $I_{121}$ and $I_{122}$. According to Eq. (5.98), they are defined as

$$\begin{aligned} I_{121}=&\frac{d \varGamma \left( \frac{d}{2}\right) }{\pi ^{d/2}Tn_2\sigma _{12}^d\chi _{12}}\left[ n_1\left( \frac{\partial \mu _1}{\partial n_1} \right) _{T,n_{2}}-T\right] \nonumber \\&-2\frac{n_1\sigma _1^d\chi _{11}}{n_2\sigma _{12}^d\chi _{12}}-\frac{n_1^2\sigma _1^d}{n_2\sigma _{12}^d\chi _{12}}\frac{\partial \chi _{11}}{\partial n_1} -\frac{n_1}{\chi _{12}}\frac{\partial \chi _{12}}{\partial n_1}, \end{aligned}$$

(5.99)

$$\begin{aligned} I_{122}=\frac{d \varGamma \left( \frac{d}{2}\right) }{\pi ^{d/2}T\sigma _{12}^d\chi _{12}} \left( \frac{\partial \mu _1}{\partial n_2} \right) _{T,n_{1}}-2- \frac{\sigma _1^d n_1}{\sigma _{12}^d\chi _{12}}\frac{\partial \chi _{11}}{\partial n_2} -\frac{n_2}{\chi _{12}}\frac{\partial \chi _{12}}{\partial n_2}. \end{aligned}$$

(5.100)

Note that for mechanically equivalent particles, $I_{121}=I_{122}=0$, which is as expected since SET and RET lead to the same Navier–Stokes transport coefficients for a monocomponent gas [27, 28].

To determine $I_{121}$ and $I_{122}$ it remains to compute the derivatives $\partial _{n_j}\chi _{ij}$ and $\partial _{n_j}\mu _i$. In the case of hard disks ($d=2$) , a good approximation for the pair distribution function $\chi _{ij}$ is given by Eq. (2.76), namely,

$$\begin{aligned} \chi _{ij}=\frac{1}{1-\phi }+\frac{9}{16}\frac{\phi }{(1-\phi )^2}\frac{\sigma _i\sigma _jM_1}{\sigma _{ij}M_2}, \end{aligned}$$

(5.101)

where $\phi =\sum _i\; n_i\pi \sigma _i^2/4$ is the solid volume fraction for disks and we recall that

$$\begin{aligned} M_\ell =\sum _{s}\; x_s \sigma _s^\ell . \end{aligned}$$

(5.102)

The expression of the chemical potential $\mu _i$ of the species i consistent with the approximation (5.101) is [38]

$$\begin{aligned} \frac{\mu _i}{T}= & {} \ln (\varXi _i^2n_i)-\ln (1-\phi )+\frac{M_1}{4M_2}\left[ \frac{9\phi }{1-\phi }+\ln (1-\phi )\right] \sigma _i \nonumber \\&-\frac{1}{8}\left[ \frac{M_1^2}{M_2^2}\frac{\phi (1-10\phi )}{(1-\phi )^2}- \frac{8}{M_2}\frac{\phi }{1-\phi }+\frac{M_1^2}{M_2^2}\ln (1-\phi )\right] \sigma _i^2, \end{aligned}$$

(5.103)

where $\varXi _i(T)$ is the (constant) thermal de Broglie wavelength of species i [39]. In the case of hard spheres ($d=3$), the pair distribution function $\chi _{ij}$ is approximated by Eq. (2.77), namely,

$$\begin{aligned} \chi _{ij}=\frac{1}{1-\phi }+\frac{3}{2}\frac{\phi }{(1-\phi )^2}\frac{\sigma _i\sigma _jM_2}{\sigma _{ij}M_3} +\frac{1}{2}\frac{\phi ^2}{(1-\phi )^3}\left( \frac{\sigma _i\sigma _jM_2}{\sigma _{ij}M_3}\right) ^2, \end{aligned}$$

(5.104)

where $\phi =\sum _i\; n_i\pi \sigma _i^3/6$ is the solid volume fraction for spheres. The chemical potential consistent with Eq. (5.104) is [39]

$$\begin{aligned} \frac{\mu _i}{T}= & {} \ln (\varXi _i^3n_i)-\ln (1-\phi )+3\frac{M_2}{M_3}\frac{\phi }{1-\phi }\sigma _i +3\left[ \frac{M_2^2}{M_3^2}\frac{\phi }{(1-\phi )^2}+ \frac{M_1}{M_3}\frac{\phi }{1-\phi }\right. \nonumber \\&\left. +\frac{M_2^2}{M_3^2}\ln (1-\phi )\right] \sigma _i^2-\left[ \frac{M_2^3}{M_3^3}\frac{\phi (2-5\phi +\phi ^2)}{(1-\phi )^3}-3 \frac{M_1M_2}{M_3^2}\frac{\phi ^2}{(1-\phi )^2}\right. \nonumber \\&\left. -\frac{1}{M_3}\frac{\phi }{1-\phi } +2\frac{M_2^3}{M_3^3}\ln (1-\phi )\right] \sigma _i^3. \end{aligned}$$

(5.105)

These results permit us to compute the quantities $I_{121}$ and $I_{122}$ in terms of the parameters of the mixture.

Appendix B

The collisional transfer contributions $D_{q,ij}^\text {c}$ and $\kappa _\text {c}$ to the heat transport coefficients are given by [15, 16]

$$\begin{aligned} D_{q,ij}^{\text {c}}= & {} \sum _{\ell =1}^{N}\frac{1}{8}\left( 1+\alpha _{i\ell }\right) m_{i\ell }\sigma _{i\ell }^{d}\chi _{i\ell }\left\{ 2B_{4} \left( 1-\alpha _{i\ell }\right) \left( \mu _{i\ell }-\mu _{\ell \,i}\right) n_{i}\left[ \frac{2}{m_\ell } D_{q,\ell j}^{\text {k}}\right. \right. \nonumber \\&\left. +\,(d+2)\frac{T_i}{T^{2}}\frac{m_{j}n_{j}}{\rho m_{i}}D_{\ell j}\right] +\frac{24B_{2}}{d+2}n_{i}\left[ \frac{2 \mu _{\ell \,i}}{m_{\ell }} D_{q,\ell j}^{\text {k}}- (d+2)\left( 2\mu _{i \ell }-\mu _{\ell \,i}\right) \right. \nonumber \\&\left. \left. \times \,\frac{T_{i}}{T^{2} }\frac{m_{j}n_{j}}{\rho m_{i}}D_{\ell j}\right] -T^{-2}\sigma _{ij}C_{i \ell j}\right\} , \end{aligned}$$

(5.106)

$$\begin{aligned} \kappa _\text {c}= & {} \sum _{i=1}^{N}\sum _{j=1}^{N}\frac{1}{8}\left( 1+\alpha _{ij}\right) m_{ij}\sigma _{ij}^{d}\chi _{ij}\left\{ 2B_{4}\left( 1-\alpha _{ij}\right) \left( \mu _{ij}-\mu _{ji}\right) n_{i} \left[ \frac{2\lambda _{j}}{m_{j}}\right. \right. \nonumber \\&\left. +\,(d+2)\frac{\rho }{m_{j}T}\left( \frac{T_j}{m_j}+\frac{T_i}{m_i}\right) D_{j}^{T}\right] +\frac{24B_{2}}{d+2}n_{i}\left[ \frac{2\lambda _j}{m_i+m_{j}}+(d+2)\frac{\rho }{T m_j}\right. \nonumber \\&\left. \left. \times \, \left( \mu _{ji}\left( \frac{T_i}{m_i}+ \frac{T_j}{m_j}\right) - 2\frac{\mu _{ij}T_{i}}{m_{i}}\right) D_{j}^{T}\right] -T^{-1}\sigma _{ij}C_{ij}\right\} , \end{aligned}$$

(5.107)

where the coefficients $B_k$ are defined by Eq. (1.177). The quantities $C_{ij}$ and $C_{i \ell j}$ are provided in terms of velocity integrals involving the zeroth-order distributions [16]. These quantities can be explicitly determined by considering the Maxwellian distributions (5.50). Their final forms can be written as

$$\begin{aligned} C_{ij}=-\frac{2\pi ^{(d-1)/2}}{d\varGamma \left( \frac{d}{2}\right) } n_{i}n_{j}\upsilon _{\text {th}}^{3} C_{ij}^*, \quad C_{i \ell j}=\frac{4\pi ^{(d-1)/2}}{d\varGamma \left( \frac{d}{2}\right) }n_in_\ell \upsilon _{\text {th}}^3C_{i\ell j}^*, \end{aligned}$$

(5.108)

where

$$\begin{aligned} C_{ij}^*= & {} (\theta _{i}+\theta _{j})^{-1/2}(\theta _{i}\theta _{j})^{-3/2}\left\{ 2\beta _{ij}^{2}+\theta _{i}\theta _{j}+(\theta _{i}+\theta _{j})\left[ (\theta _{i}+\theta _{j})\mu _{ij}\mu _{ji}\right. \right. \nonumber \\&\left. \left. +\beta _{ij}(1+\mu _{ji})\right] \right\} +\,\frac{3}{4}(1-\alpha _{ij})(\mu _{ji}-\mu _{ij})\left( \frac{\theta _{i}+\theta _{j}}{\theta _{i}\theta _{j}}\right) ^{3/2}\nonumber \\&\times \left[ \mu _{ji}+\beta _{ij}(\theta _{i}+\theta _{j})^{-1}\right] , \end{aligned}$$

(5.109)

$$\begin{aligned} C_{i \ell j}^*= & {} (\theta _i+\theta _\ell )^{-1/2}(\theta _i\theta _\ell )^{-3/2}\Big \{ \delta _{j\ell }\beta _{i\ell }(\theta _i+\theta _\ell ) -\frac{1}{2}\theta _i\theta _\ell \nonumber \\&\times \left[ 1+\frac{\mu _{\ell \,i}(\theta _i+\theta _\ell )-2 \beta _{i\ell }}{\theta _\ell }\right] \frac{\partial \ln \gamma _\ell }{\partial \ln n_j}\Big \}+\frac{1}{4}(1-\alpha _{i \ell })(\mu _{\ell \, i}-\mu _{i\ell })\nonumber \\&\times \left( \frac{\theta _i+\theta _\ell }{\theta _i\theta _\ell }\right) ^{3/2} \left( \delta _{j\ell } +\frac{3}{2}\frac{\theta _i}{\theta _i+\theta _\ell }\frac{\partial \ln \gamma _\ell }{\partial \ln n_j}\right) , \end{aligned}$$

(5.110)

where we recall that $\beta _{ij}=\mu _{ij}\theta _j-\mu _{ji}\theta _i$.

The kinetic contributions $D_{q,ij}^\text {k}$ and $\kappa _\text {k}$ for a binary mixture ($N=2$) are now considered. They are defined by Eq. (5.72). To determine them, we must also evaluate the parameters $\lambda _i$ and $d_{q,ij}$. These parameters can be expressed in a more compact form by introducing their dimensionless expressions

$$\begin{aligned} d_{q,ij}^*=\frac{2}{d+2}\frac{(m_1+m_2)\nu '}{n}d_{q,ij}, \quad \lambda _{i}^*=\frac{2}{d+2}\frac{(m_1+m_2)\nu '}{n T}\lambda _{i}. \end{aligned}$$

(5.111)

The coefficients $\lambda _i^*$ are [32]

$$\begin{aligned} \lambda _1^*=\frac{(\psi _{22}^*-2\zeta _0^*)\overline{\lambda }_1^*-\psi _{12}^*\overline{\lambda }_2^*}{4\zeta _0^{*2}-2(\psi _{11}^*+\psi _{22}^*)\zeta _0^*-\psi _{12}^*\psi _{21}^*+\psi _{11}^*\psi _{22}^*}, \end{aligned}$$

(5.112)

$$\begin{aligned} \lambda _2^*=\frac{(\psi _{11}^*-2\zeta _0^*)\overline{\lambda }_2^*-\psi _{21}^*\overline{\lambda }_1^*}{4\zeta _0^{*2}-2(\psi _{11}^*+\psi _{22}^*)\zeta _0^*-\psi _{12}^*\psi _{21}^*+\psi _{11}^*\psi _{22}^*}, \end{aligned}$$

(5.113)

where $\zeta _0^*= \zeta ^{(0)}/\nu '$ can be easily obtained from Eq. (5.51) and

$$\begin{aligned} \overline{\lambda }_i^*= & {} \frac{m_1+m_2}{m_i}x_i\gamma _i^2\sum _{j=1}^2\left( \delta _{ij}-\frac{\omega _{ij}^*- \zeta _0^*\delta _{ij}}{x_j\gamma _j}D_j^{*T}+\frac{\pi ^{d/2}}{d(d+2)\varGamma \left( \frac{d}{2}\right) } n\sigma _{ij}^d \mu _{ij}x_j \chi _{ij}\right. \nonumber \\&\times \left. \frac{\gamma _j (1+\alpha _{ij})A_{ij}}{\gamma _i}\right) . \end{aligned}$$

(5.114)

Here, $A_{ij}$ is defined as

$$\begin{aligned} A_{ij}= & {} (d+2)(\mu _{ij}^{2}-1)+(2d-5-9\alpha _{ij})\mu _{ij}\mu _{ji}+\mu _{ji}^{2} \nonumber \\\times & {} (d-1+3\alpha _{ij}+6\alpha _{ij}^{2}) +\frac{6\mu _{ji}^2(1+\alpha _{ij})^{2}\theta _{i}}{\theta _j}, \end{aligned}$$

(5.115)

and the reduced collision frequencies $\omega _{ij}^*$ and $\psi _{ij}^*$ are given by

$$\begin{aligned} \omega _{11}^*= & {} \frac{\pi ^{(d-1)/2}}{\varGamma \left( \frac{d}{2}\right) }\frac{2}{d\sqrt{2}} \left( \frac{\sigma _1}{\sigma _{12}}\right) ^{d-1}x_1 \chi _{11} \theta _1^{-1/2}(1-\alpha _{11}^2)\nonumber \\&+\frac{\pi ^{(d-1)/2}}{\varGamma \left( \frac{d}{2}\right) }\frac{2}{d(d+2)} x_2\chi _{12}\mu _{21}(1+\alpha _{12})\nonumber \\&\times (\theta _1+\theta _2)^{-1/2}\theta _1^{1/2} \theta _2^{-3/2} F, \end{aligned}$$

(5.116)

$$\begin{aligned} \omega _{12}^*=\frac{\pi ^{(d-1)/2}}{\varGamma \left( \frac{d}{2}\right) }\frac{2}{d(d+2)} x_2\chi _{12}\mu _{21}(1+\alpha _{12})(\theta _1+\theta _2)^{-1/2}\theta _1^{1/2} \theta _2^{-3/2}G, \end{aligned}$$

(5.117)

$$\begin{aligned} \psi _{11}^*= & {} \frac{\pi ^{(d-1)/2}}{\varGamma \left( \frac{d}{2}\right) } \frac{8}{d(d+2)}\left( \frac{\sigma _1}{\sigma _{12}}\right) ^{d-1} x_1 \chi _{11}(2\theta _1)^{-1/2} (1+\alpha _{11})\left[ \frac{d-1}{2}\right. \nonumber \\&\left. +\frac{3}{16}(d+8)(1-\alpha _{11})\right] +\frac{\pi ^{(d-1)/2}}{\varGamma \left( \frac{d}{2}\right) } \frac{1}{d(d+2)}x_2\chi _{12}\mu _{21}(1+\alpha _{12})\nonumber \\&\times \left( \frac{\theta _1}{\theta _2(\theta _1+\theta _2)}\right) ^{3/2}\left[ H-(d+2)\frac{\theta _1+\theta _2}{\theta _1}F\right] , \end{aligned}$$

(5.118)

$$\begin{aligned} \psi _{12}^*=&-\frac{\pi ^{(d-1)/2}}{\varGamma \left( \frac{d}{2}\right) } \frac{1}{d(d+2)}x_1\chi _{12}\mu _{12}(1+\alpha _{12})\left( \frac{\theta _2}{\theta _1(\theta _1+\theta _2)}\right) ^{3/2}\nonumber \\&\times \left[ I+(d+2)\frac{\theta _1+\theta _2}{\theta _2}G\right] . \end{aligned}$$

(5.119)

In the above equations, the following quantities have been introduced

$$\begin{aligned} F= & {} (d+2)(2\beta _{12}+\theta _2)+\mu _{21}(\theta _1+\theta _2)\bigg \{(d+2)(1-\alpha _{12}) -[(11+d)\alpha _{12}\nonumber \\&-5d-7]\times \,\beta _{12}\theta _1^{-1}\bigg \} +3(d+3)\beta _{12}^2\theta _1^{-1}+2\mu _{21}^2\nonumber \\&\times \left( 2\alpha _{12}^{2}-\frac{d+3}{2}\alpha _{12}+d+1\right) \theta _1^{-1}(\theta _1+\theta _2)^2\nonumber \\&-\, (d+2)\theta _2\theta _1^{-1}(\theta _1+\theta _2), \end{aligned}$$

(5.120)

$$\begin{aligned} G= & {} (d+2)(2\beta _{12}-\theta _1)+\mu _{21}(\theta _1+\theta _2)\bigg \{(d+2)(1-\alpha _{12}) +[(11+d)\alpha _{12}\nonumber \\&-5d-7] \times \,\beta _{12}\theta _2^{-1}\bigg \} -3(d+3)\beta _{12}^2\theta _2^{-1}-2\mu _{21}^2\nonumber \\&\times \left( 2\alpha _{12}^{2}-\frac{d+3}{2}\alpha _{12}+d+1\right) \theta _2^{-1}(\theta _1+\theta _2)^2\nonumber \\&+\, (d+2)(\theta _1+\theta _2), \end{aligned}$$

(5.121)

$$\begin{aligned} H= & {} 2\mu _{21}^2\theta _1^{-2}(\theta _1+\theta _2)^2 \left( 2\alpha _{12}^{2}-\frac{d+3}{2}\alpha _{12}+d+1\right) \big [(d+2)\theta _1+(d+5)\theta _2\big ]\nonumber \\&-\,\mu _{21}(\theta _1+\theta _2) \bigg \{\beta _{12}\theta _1^{-2}[(d+2)\theta _1+(d+5)\theta _2][(11+d)\alpha _{12} -5d-7]\nonumber \\&-\,\theta _2\theta _1^{-1}[20+d(15-7\alpha _{12})+d^2(1-\alpha _{12})-28\alpha _{12}] -(d+2)^2(1-\alpha _{12})\bigg \} \nonumber \\&+\,3(d+3)\beta _{12}^2\theta _1^{-2}[(d+2)\theta _1+(d+5)\theta _2]+ 2\beta _{12}\theta _1^{-1} \left[ (d+2)^2\theta _1\right. \nonumber \\&\left. +\,(24+11d+d^2)\theta _2\right] +(d+2)\theta _2\theta _1^{-1} [(d+8)\theta _1+(d+3)\theta _2]\nonumber \\&-\,(d+2)(\theta _1+\theta _2)\theta _1^{-2}\theta _2 [(d+2)\theta _1+(d+3)\theta _2], \end{aligned}$$

(5.122)

$$\begin{aligned} I= & {} 2\mu _{21}^2\theta _2^{-2}(\theta _1+\theta _2)^2 \left( 2\alpha _{12}^{2}-\frac{d+3}{2}\alpha _{12}+d+1\right) \big [(d+5)\theta _1+(d+2)\theta _2\big ]\nonumber \\&-\,\mu _{21}(\theta _1+\theta _2) \bigg \{\beta _{12}\theta _2^{-2}[(d+5)\theta _1+(d+2)\theta _2][(11+d)\alpha _{12} -5d-7]\nonumber \\&+\,\theta _1\theta _2^{-1}[20+d(15-7\alpha _{12})+d^2(1-\alpha _{12})-28\alpha _{12}] +(d+2)^2(1-\alpha _{12})\bigg \} \nonumber \\&+\,3(d+3)\beta _{12}^2\theta _2^{-2}[(d+5)\theta _1+(d+2)\theta _2]- 2\beta _{12}\theta _2^{-1}\big [(24+11d+d^2)\theta _1 \nonumber \\&+\,(d+2)^2\theta _2\big ]+(d+2)\theta _1\theta _2^{-1} [(d+3)\theta _1+(d+8)\theta _2]\nonumber \\&-\,(d+2)(\theta _1+\theta _2)\theta _2^{-1} [(d+3)\theta _1+(d+2)\theta _2]. \end{aligned}$$

(5.123)

The expressions for $\omega _{22}^*$, $\omega _{21}^*$, $\psi _{22}^*$, and $\psi _{21}^*$ can be easily obtained from Eqs. (5.116)–(5.119) by interchanging $1\leftrightarrow 2$. With these results the (scaled) kinetic coefficient

$$\begin{aligned} \kappa _\text {k}^{*}= \frac{2}{d+2}\frac{(m_1+m_2)\nu '}{n T}\kappa _\text {k} \end{aligned}$$

(5.124)

can be finally written as

$$\begin{aligned} \kappa _\text {k}^{*}=\frac{\overline{\lambda }_1^*(\psi _{22}^*-2\zeta _0^*-\psi _{21}^*)+ \overline{\lambda }_2^*(\psi _{11}^*-2\zeta _0^*-\psi _{12}^*)}{4\zeta _0^{*2}-2(\psi _{11}^*+\psi _{22}^*)\zeta _0^*-\psi _{12}^*\psi _{21}^*+\psi _{11}^*\psi _{22}^*} +\left( \frac{\gamma _1}{\mu _{12}}-\frac{\gamma _2}{\mu _{21}}\right) D_1^{T*}, \end{aligned}$$

(5.125)

where $D_1^{T*}= (\rho \nu '/n T)D_1^T$ and use has been made of Eqs. (5.112) and (5.113).

The (reduced) coefficients $d_{q,ij}^*$ are now considered. In a binary mixture, the relevant coefficients are the set $\left\{ d_{q,11}^*, d_{q,12}^*, d_{q,22}^*, d_{q,21}^*\right\}$. The explicit expressions of $d_{q,11}^*$ and $d_{q,12}^*$ are [32]

$$\begin{aligned} d_{q,11}^*=\frac{\overline{d}_{q,21}^*\psi _{12}^*-\overline{d}_{q,11}^*(\psi _{22}^*-\frac{3}{2}\zeta _0^*)}{\psi _{12}^*\psi _{21}^*+(\psi _{11}^*-\frac{3}{2}\zeta _0^*)(\frac{3}{2}\zeta _0^*-\psi _{22}^*)}, \end{aligned}$$

(5.126)

$$\begin{aligned} d_{q,12}^*=\frac{\overline{d}_{q,22}^*\psi _{12}^*-\overline{d}_{q,12}^*(\psi _{22}^*-\frac{3}{2}\zeta _0^*)}{\psi _{12}^*\psi _{21}^*+(\psi _{11}^*-\frac{3}{2}\zeta _0^*)(\frac{3}{2}\zeta _0^*-\psi _{22}^*)}, \end{aligned}$$

(5.127)

where the (dimensionless) coefficients $\overline{d}_{q,ij}^*$ are given by

$$\begin{aligned} \overline{d}_{q,ij}^*= & {} \frac{m_1+m_2}{m_i}x_i\gamma _in_j\frac{\partial \gamma _i}{\partial n_j}- \frac{m_1+m_2}{m_i}x_ix_j\gamma _i^2\sum _{\ell =1}^2\frac{\omega _{i\ell }^*- \zeta _0^*\delta _{i\ell }}{x_\ell \gamma _\ell }D_{\ell j}^{*} +\frac{n_j}{\nu '}\frac{\partial \zeta ^{(0)}}{\partial n_j}\lambda _i^* \nonumber \\&+\,\frac{\pi ^{d/2}}{d(d+2)\varGamma \left( \frac{d}{2}\right) } \frac{m_1+m_2}{m_i}x_i\gamma _i^2n \sum _{\ell =1}^2 \mu _{\ell \, i}\;x_\ell \;\sigma _{i\ell }^d\;\chi _{i\ell }(1+\alpha _{i\ell }) \nonumber \\&\times \, \left\{ \left[ \delta _{j\ell }+\frac{1}{2} \left( n_j\frac{\partial \ln \chi _{i\ell }}{\partial n_j}+I_{i\ell j}\right) \right] B_{i\ell }+ \frac{\theta _i}{\theta _{\ell }}n_j\frac{\partial \ln \gamma _\ell }{\partial n_j}A_{i\ell }\right\} . \end{aligned}$$

(5.128)

In Eq. (5.127), $A_{ij}$ is defined by Eq. (5.115) while in Eq. (5.128) $B_{ij}$ is

$$\begin{aligned} B_{ij}= & {} (d+8)\mu _{ij}^{2}+(7+2d-9\alpha _{ij})\mu _{ij}\mu _{ji}+(2+d+3\alpha _{ij}^{2}-3\alpha _{ij})\mu _{ji}^{2}\nonumber \\&+\,3\mu _{ji}^{2}(1+\alpha _{ij})^{2}\frac{\theta _{i}^2}{\theta _j^2}+\left[ (d+2)\mu _{ij}^{2}+(2d-5-9\alpha _{ij})\mu _{ij}\mu _{ji}\right. \nonumber \\&\left. +\,(d-1+3\alpha _{ij}+6\alpha _{ij}^{2})\mu _{ji}^{2}\right] \frac{\theta _{i}}{\theta _j}-(d+2)\frac{\theta _i+\theta _j}{\theta _j}. \end{aligned}$$

(5.129)

The expressions of the coefficients $d_{q,22}^*$ and $d_{q,21}^*$ can be derived from Eqs. (5.126) and (5.127), respectively, by interchanging $1\leftrightarrow 2$. With these results the kinetic coefficients $D_{q,ij}$ can be obtained from the second relation of Eq. (5.72).

The reduced forms of the collisional contributions to $\kappa$ and $D_{q,ij}$ can be determined from Eqs. (5.106) and (5.107), respectively. In the case of a binary mixture, the expressions of $\kappa _\text {c}^{*}= (2/(d+2))[(m_1+m_2)\nu '/nT]\kappa _\text {c}$ and $D_{q,ij}^{c*}= (2/(d+2))[(m_1+m_2)\nu '/n]D_{q,ij}^\text {c}$ are

$$\begin{aligned} \kappa _\text {c}^{*}= & {} \frac{3}{2}\frac{\pi ^{d/2}}{d(d+2)\varGamma \left( \frac{d}{2}\right) }n^* \sum _{i=1}^2\sum _{j=1}^2 x_i\left( \frac{\sigma _{ij}}{\sigma _{12}}\right) ^d\chi _{ij}\mu _{ij}(1+\alpha _{ij})\Bigg \{\bigg [(1-\alpha _{ij})\mu _{ij} \nonumber \\&+\,(3+\alpha _{ij}) \mu _{ji}\bigg ]\lambda _{j}^* +(m_1+m_2)D_j^{T*}\bigg [\frac{\gamma _j}{m_j}\bigg ((1-\alpha _{ij})\mu _{ij}+(3+\alpha _{ij})\mu _{ji}\bigg ) \nonumber \\&+\,\frac{\gamma _i}{m_i}\bigg ((3+\alpha _{ij})\mu _{ji}-(7+\alpha _{ij})\mu _{ij}\bigg )\bigg ]\nonumber \\&+\frac{16}{3\sqrt{\pi }} \frac{x_jm_j}{m_1+m_2}n^*\left( \frac{\sigma _{ij}}{\sigma _{12}}\right) C_{ij}^*\Bigg \}, \end{aligned}$$

(5.130)

$$\begin{aligned} D_{q,ij}^{c*}= & {} \frac{3}{2}\frac{\pi ^{d/2}}{d(d+2)\varGamma \left( \frac{d}{2}\right) }x_i n^*\sum _{\ell =1}^2 \left( \frac{\sigma _{i\ell }}{\sigma _{12}}\right) ^d\chi _{i\ell }\mu _{i\ell }(1+\alpha _{i\ell }) \Bigg \{\bigg [(1-\alpha _{i\ell })\mu _{i\ell } \nonumber \\&+\,(3+\alpha _{i\ell })\mu _{\ell \,i}\bigg ] d_{q,\ell j}^* +(m_1+m_2)x_jD_{\ell j}^*\bigg [\frac{\gamma _\ell }{m_\ell }\bigg ((1-\alpha _{i\ell })\mu _{i\ell } +(3+\alpha _{i\ell })\mu _{\ell \,i}\bigg ) \nonumber \\&+\,\frac{\gamma _i}{m_i}\bigg ((3+\alpha _{i\ell })\mu _{\ell \,i} -(7+\alpha _{i\ell })\mu _{i\ell }\bigg )\bigg ] -\frac{32}{3\sqrt{\pi }}\frac{x_\ell m_\ell }{m_1+m_2}n^*\left( \frac{\sigma _{i\ell }}{\sigma _{12}}\right) C_{i\ell j}^*\Bigg \},\nonumber \\ \end{aligned}$$

(5.131)

where $n^*= n\sigma _{12}^d$.